Confusion over Distribution

Hi,

Hopefully someone can enlighten me:
I have a collection of data (output from a computer simulation) where one hundred simulations were run to find the number of times a task had to be completed before it was successful. The result is 100 numbers each giving the number of times attempted before the task was successful. I believe this data is discrete numerical data. I have run this with one variable altered, which affects the number of attempts before success is achieved. I want to establish whether the differences between the number of attempts before success for each different value for my variable is significant. The problem is, I don't believe the data to be a normal distribution since the median doesn't equal the mean and the graph is not in any way bell curved. It's not binomial because the probability of success increases with each attempt. So how can I figure out what kind of distribution it is and thus what statistical tests to do to establish significance?